Determine whether the following operations define a binary operat… - Vidyadip

Question

Determine whether the following operations define a binary operation on the given set or not:
$'+6\ '$ on $S = {0, 1, 2, 3, 4, 5}$ defined by $, \text{a}+_6\text{b}=\begin{cases}\text{a}+\text{b},&\text{if a}+\text{b}<6\\\text{a}+\text{b}-6,&\text{if a}+\text{b}\geq6\end{cases}$

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Answer

We have $, S = {0, 1, 2, 3, 4, 5}$ and $, \text{a}+_6\text{b}=\begin{cases}\text{a}+\text{b},&\text{if a}+\text{b}<6\\\text{a}+\text{b}-6,&\text{if a}+\text{b}\geq6\end{cases}$
Let $\text{a}\in\text{S}$ and $\text{b}\in\text{S}$ such that $a + b < 6$ Then $\text{a}+_6\text{b}=\text{a}+\text{b}\in\text{S}$ $\big[\because a + b < 6 = 0, 1, 2, 3, 4, 5 \big]$
Let $\text{a}\in\text{S}$ and $\text{b}\in\text{S}$ such that $a + b > 6$ Then $\text{a}+_6\text{b}=\text{a}+\text{b}-6\in\text{S}$ $\big[\because\ \text{if a}+\text{b}\geq6$ then $\text{a}+\text{b}-6\geq6 = 0, 1, 2, 3, 4, 5 \big]$
$\therefore\ \text{a}+_6\text{b}\in\text{S}$ for $\text{a, b}\in\text{S}$
$\therefore +_6$ defined a binary operation on $S$.

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